由几乎收敛的二元展开序列定义的集合的Hausdorff维数

Pub Date : 2023-03-13 DOI:10.1017/S0017089523000046

Q. Song

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引用次数: 0

摘要

摘要本文研究了由几乎收敛的二元展开序列定义的集合的Hausdorff维数。更准确地说，对于[0,1]$中的任何$\alpha\，都可以确定以下集合\beggin｛align*｝\bigg｛x\ in[0,1）\；：\；\frac｛1｝｛n｝\sum_｛k=a｝^｛a+n-1｝x_｛k｝\longrightarrow\alpha\textrm的Hausdorff维数。这就完成了Usachev[Glasg.Math.J.64（2022），691–697]考虑的一个问题，其中只给出了有理$\alpha$的维数。

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Hausdorff dimension of sets defined by almost convergent binary expansion sequences

Abstract In this paper, we study the Hausdorff dimension of sets defined by almost convergent binary expansion sequences. More precisely, the Hausdorff dimension of the following set \begin{align*} \bigg\{x\in[0,1)\;:\;\frac{1}{n}\sum_{k=a}^{a+n-1}x_{k}\longrightarrow\alpha\textrm{ uniformly in }a\in\mathbb{N}\textrm{ as }n\rightarrow\infty\bigg\} \end{align*} is determined for any $ \alpha\in[0,1] $ . This completes a question considered by Usachev [Glasg. Math. J. 64 (2022), 691–697] where only the dimension for rational $ \alpha $ is given.

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